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In this thesis we develop numerical algorithms for stochastic control problems where the state processes is an Itˆo process that depends on the control process in both the drift and diffusion functions. We derive sufficient conditions for convergence of a proximal policy gradient method (PPGM) for the stochastic linear quadratic control problem. The optimal control can be uniquely determined by solving a Ricatti equation. We prove that the control induced by the policy gradient method converges to the optimal control. Considering convergence analysis in the general case, we study the underlying backwards stochastic differential equations using Malliavin Calculus, and determine some conditions under which convergence of the control process can be established. To implement the PPGM method, we use concepts in machine learning methods, and extend our algorithms using duality theory. Using deep learning methods, the algorithms developed are scalable to high dimensional problems, with a reasonable runtime. Duality allows us to solve an auxiliary control problem simultaneously to the original primal problem, which allows us to exploit the dual-primal relations between the associated processes, and form tight bounds of the value function. In certain cases solving the dual problem directly bypasses complexity within the primal problem. We further use duality to implement algorithms that can be applied to non-Markovian control problems.

We study a few constrained Stochastic Optimal Control Problems. First, we look at problems with terminal constraints. For various convex problems with constrained control, such as Linear Quadratic Mean-Field problem or Non-Markovian problem with stochastic coefficients, we draw equivalence relationship between the Fritz John condition and Karush-Kuhn-Tucker (KKT) conditions. Then we construct an unconstrained problem with the Lagrange Multiplier derived from Fritz John condition. Finally, we show the equivalence between the optimality of the unconstrained problem and its original problem. Furthermore, we look at the Duality of Linear Quadratic Mean-Field control problems and find an equivalence relationship between the primal and dual problems in the absence of control constraints. Lastly we compare the Riccati solutions to the Linear Quadratic Mean-Field control problem and the empirical solutions to the Mean-Field Forward Backward Stochastic Differential Equations (FBSDEs) using Deep Learning to verify our results.

In this thesis we study stochastic control problems with control-dependent stopping terminal time. We assess what are the methods and theorems from standard control optimization settings that can be applied to this framework and we introduce new statements where necessary. In the first part of the thesis we study a general optimal liquidation problem with a control-dependent stopping time which is the first time the stock holding becomes zero or a fixed terminal time, whichever comes first. We prove a stochastic maximum principle (SMP) which is markedly different in its Hamiltonian condition from that of the standard SMP with fixed terminal time. The new version of the SMP involves an innovative definition of the FBSDE associated to the problem and a new type of Hamiltonian. We present several examples in which the optimal solution satisfies the SMP in this thesis but fails the standard SMP in the literature. The generalised version of the SMP Theorem can also be applied to any problem in physics and engineering in which the terminal time of the optimization depends on the control, such as optimal planning problems. In the second part of thesis, we introduce an optimal liquidation problem with control-dependent stopping time as before. We analyze the case when an agent is trading on a market with two financial assets correlated with each other. The agent's task is to liquidate via market orders an initial position of shares of one of the two financial assets, without having the possi- bility of trading the other stock. The main results of this part consist in proving a verification theorem and a comparison principle for the viscosity solution to the HJB equation and finding an approximation of the classical solution of the Hamilton-Jacobi-Bellman (HJB) equation associated to this problem.

We present a mathematical framework for modelling the dynamics of limit order books, built on the combination of two modelling ingredients: the order flow -modelled as a general spatial point process- and market clearing, modelled via a deterministic 'mass transport' operator acting on distributions of buy and sell orders. At the mathematical level, this corresponds to a natural decomposition of the infinitesimal generator describing the evolution of the limit order book into two operators: the generator of the order flow and the clearing operator. Our model provides a flexible framework for modelling and simulating order book dynamics and studying various scaling limits of discrete order book models. We show that our framework includes previous models as special cases and yields insights into the interaction between the order flow and price dynamics, the use of order book data for prediction of intraday price movements. The framework also allows for model comparison and the study of the order flow. The modular structure of the model is well-adapted to simulation and allows the stochastic model for the order flow and the clearing mechanism to be specified independently. Then, as a simple demonstration, models with different assumptions on the order intensities are compared. The simulation result shows that orders of relatively large size also play an essential role in the evolution of the order book process. We further investigate the asymptotic behaviour of the order book processes, including the fluid scaling and the diffusion scaling. The decomposition relation between the order flow and the order book holds when the ask and bid price do not move. In general, the price processes depend on the order flow and the current state of the order book. We prove that as the tick size becomes small, the ask price and bid price converge to the same limiting process.

Competition between market makers, which considers the impacts on trading strat egy of individual and liquidity of whole market resulting from multiple market mak ers competing for order flow and market maker incentives, was not properly studied in the literature of optimal market making problem. This thesis is devoted to the optimal market making problem, with competition between market makers. Three main topics are studied in this thesis. In the first topic, we consider the price competition between market makers. We discuss optimal market marking with price competition and incomplete information, which results in a looping dependence structure among market makers. We solve the problem with the non-zero-sum stochastic differential game approach and char acterize the equilibrium value function with a coupled system of nonlinear ordinary differential equations. We prove, do not assume a priori, that the Issac condition is satisfied, which ensures the existence and uniqueness of Nash equilibrium. We also perform some numerical tests that show our model produces tighter bid/ask spread than a benchmark model without price competition and improves market liquidity. In the second topic, we consider market makers competing for the market maker incentive reward proposed by exchange, which depends on their trading volume ranking. We model the competition as a stochastic mean field game, which can be further reduced to a finite state mean field game, whose equilibrium is characterized by a forward backward ODE systems. We numerically solve the equilibrium with the deep neural network approach proposed in our third topic, and perform some numerical tests to compare bid/ask spread under different types market maker in centive reward. It is suggested that the introduction of incentive can reduce the implicit trading cost, and rank-based reward, compared with the linear trading vol ume reward, can produce lower best bid/ask spread. In the third topic, we discuss the deep neural network approach for solving the forward backward ODE system corresponding to a more general class of finite state mean field game, and the game in the second topic is just a special case of it. We prove that the error between true solution and our approximation is linear to the square root of loss function of our deep neural network.

Credit risk, which considers the risk of loss resulting from a counterparty's failure to meet contractual obligations, was not properly studied in the literature of dynamic portfolio optimization problem until the 2008-2009 financial crisis. This thesis is devoted to the dynamic portfolio optimization with credit risk. Two main topics are studied in this thesis. In the first topic, we consider a utility maximization problem with defaultable stocks and looping contagion risk. We assume that the default intensity of one company depends on the stock prices of itself and other companies, and the default of the company induces immediate drops in the stock prices of the surviving companies. Under such looping contagion risk framework, we prove that the value function is the unique viscosity solution of the HJB equation. We also perform some numerical tests to compare and analyse the statistical distributions of the terminal wealth of log utility and power utility based on two strategies, one using the full information of intensity process and the other a proxy constant intensity process. The numerical tests confirm that modeling looping contagion risk properly is important, especially in financial distressed period. The second topic is on dynamic portfolio optimization with contingent convertible (CoCo) bond. As a new type of hybrid product, CoCo bond has the interesting feature that converting from debt to equity is contingent. We model the conversion of CoCo bond by reduced-form approach and assume that the conversion intensity is a deterministic function of the coupon rate and the issuing bank's stock price. Theoretically, we construct the viscosity solution representation between the value function and the corresponding HJB equation. Practically, we compare the performance between investing into CoCo bond and the issuing bank's stock. We analyse the statistical distributions of terminal wealth of log utility and power utility based on these two investment choices. Our simulation results show that, the CoCo bond holders bear more loss than equity holders if conversion occurs. However, investing into CoCo bond gets more profit (mean) while bearing less market risk (volatility) as long as conversion does not occur.

This thesis seeks to gain further insight into the connection between stochastic optimal control and forward and backward stochastic differential equations and its applications in solving continuous-time constrained portfolio optimization problems. Three topics are studied in this thesis. In the first part of the thesis, we focus on stochastic maximum principle, which seeks to establish the connection between stochastic optimal control and backward stochastic differential differential equations coupled with static optimality condition on the Hamiltonian. We prove a weak neccessary and sufficient maximum principle for Markovian regime switching stochastic optimal control problems. Instead of insisting on the maxi- mum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarkes generalized gradient of the Hamiltonian and Clarkes normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle. In the second part of the thesis, we study a continuous-time stochastic linear quadratic control problem arising from mathematical finance. We model the asset dynamics with random market coefficients and portfolio strategies with convex constraints. Following the convex duality approach,we show that the necessary and sufficient optimality conditions for both the primal and dual problems can be written in terms of processes satisfying a system of FBSDEs together with other conditions. We characterise explicitly the optimal wealth and portfolio processes as functions of adjoint processes from the dual FBSDEs in a dynamic fashion and vice versa. We apply the results to solve quadratic risk minimization problems with cone-constraints and derive the explicit representations of solutions to the extended stochastic Riccati equations for such problems. In the final section of the thesis, we extend the previous result to utility maximization problems. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of FBSDEs plus additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint processes coming from the dual FBSDEs in a dynamic fashion and vice versa. Moreover, we also find that the optimal primal wealth process coincides with the optimal adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems and contrasts the simplicity of the duality approach we propose with the technical complexity in solving the primal problem directly.

In this thesis, two topics in portfolio management have been studied: utility-risk portfolio selection and a paradox in time consistency in mean-variance problem. The first topic is a comprehensive study on utility maximization subject to deviation risk constraints. Under the complete Black-Scholes framework, by using the martingale approach and mean-field heuristic, a static problem including a variational inequality and some constraints on nonlinear moments, called Nonlinear Moment Problem, has been obtained to completely characterize the optimal terminal payoff. By solving the Nonlinear Moment Problem, the various well-posed mean-risk problems already known in the literature have been revisited, and also the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems has been established under the assumption that the underlying utility satisfies the Inada Condition. To the best of our knowledge, the positive answers to the latter two problems have long been absent in the literature. In particular, the existence of an optimal solution for utility-semivariance problem, an example of the utility-downside-risk problem, is in substantial contrast to the nonexistence of an optimal solution for the mean-semivariance problem. This existence result allows us to utilize semivariance as a risk measure in portfolio management. Furthermore, it has been shown that the continuity of the optimal terminal wealth in pricing kernel, thus the solutions in the binomial tree models converge to the solution in the continuous-time Black-Scholes model. The convergence can be applied to provide a numerical method to compute the optimal solution for utility-deviation-risk problem by using the optimal portfolios in the binomial tree models, which are easily computed; such numerical algorithm for optimal solution to utility-risk problem has been absent in the literature. In the second part of this thesis, a paradox in time consistency in mean-variance has been established. People often change their preference over time, so the maximizer for current preference may not be optimal in the future. We call this phenomenon time inconsistency or dynamic inconsistency. To manage the issues of time inconsistency, a game-theoretic approach is widely utilized to provide a time-consistent equilibrium solution for dynamic optimization problem. It has been established that, if investors with mean-variance preference adopt the equilibrium solutions, an investor facing short-selling prohibition can acquire a greater objective value than his counterpart without the prohibition in a buoyant market. It has been further shown that the pure strategy of solely investing in bond can sometimes simultaneously dominate both constrained and unconstrained equilibrium strategies. With numerical experiments, the constrained investor can dominate the unconstrained one for more than 90% of the time horizon. The source of paradox is rooted from the nature of game-theoretic approach on time consistency, which purposely seeks for an equilibrium solution but not the ultimate maximizer. Our obtained results actually advocate that, to properly implement the concept of time consistency in various financial problems, all economic aspects should be critically taken into account at a time.

This thesis is devoted to the asset allocation and portfolio optimization with small transaction costs. Three topics are studied. The first and second topics are on asset allocation problems with purely proportional transaction cost and strictly positive transaction cost, respectively. The fundamental objective is to keep the asset portfolio close to a target portfolio and in the meantime to reduce the trading cost in doing so. For each problem, we derive the quasi-variational inequality and prove a verification theorem giving sufficient conditions for a QVI solution to be the value function. The optimal strategy is a singular control in the first topic and an impulse control in the second. Furthermore, we provide a matrix exponential representation of the QVI solution for both problems and perform asymptotic analysis to characterize the optimal no transaction region when transaction costs are sufficiently small. Additionally, for both topics, we apply the asymptotic results for the boundary points and derive an expansion for the QVI solution, the optimality of which can be shown via verification theorem (up to leading orders). The third topic is on portfolio optimization with proportional transaction cost. We construct an efficient frontier problem (EFP) of maximizing expected terminal utility and minimizing terminal CVAR. We first solve the frictionless case by duality approach and nonsmooth analysis. For three representative utility functions, we obtain numerically the optimal trading strategy, optimal expected terminal utility and optimal CVAR. Our analysis of how these three quantities change with respect to different CVAR constraints provides flexibility for an investor to balance return and risk according to her own preference. We then include transaction cost to the EFP, which is equivalent to including transaction cost to a non-smooth utility maximization problem. Asymptotic analysis gives expansions for no transaction boundaries which are then applied to EFPs with different utility functions. This topic ends by numerical analysis on impact of introducing CVAR constraint and/or transaction cost for classical utility maximization problem.

In this thesis, we study two classes of point processes by analysing key properties and discussing applications in finance and insurance. The first point process studied was the default indicator process in credit risk modelling. We considered a pure jump Lévy process of finite variation for the asset value and an unobservable random barrier. The default time was defined as the first time the asset value falls below the barrier. Using the indistinguishable intensity process and the instantaneous likelihood process, we proved the absolute continuity of the compensator for the default indicator process, or equivalently, the existence of the intensity process of the default time. Moreover, we found the explicit representation of the intensity in terms of the distance between the asset value and its running minimal value, thus the intensity is an endogenous process, which sheds new light on the relationship between the intensity model and the structural model. The second class of point processes is the Dynamic Contagion Process, which has intensities modelled with a shot-noise component describing the external impact and mutually-exciting jump components that describe the internal contagion effect. In the bivariate case, we found the stationarity condition with which we explored the diffusion approximation of the high frequency point process system and applied it in filtering. In the univariate case, we constructed a pure jump process derived from a dynamic contagion process and showed the weak convergence to a Cox-Ingersoll- Ross model (CIR) process. The pathwise approximation provides an alternative method of simulating the square-root processes and can be further extended to the approximation of the Heston model in option pricing.

In this thesis, we discuss the optimal liquidation problem in a finite horizon model with permanent and temporary pricing impact. We use different model set-ups including a finite time Markov diffusion model and a regime switching model with exit time. The drift and diffusion terms of the asset price are general functions depending on the state variables as well as the control. There is also a non-linear transaction cost associated with the liquidation. We verify the continuity of the value function and show that it is the unique viscosity solution of the associated HJB equation. We also propose a perturbation method to approximate the viscosity solution through a series of classical solutions with the help of the stability property of viscosity solutions. We revise the definition of viscosity solutions for the regime switching model and show that the value function is a strong-form viscosity solution. Numerical results are presented at the end to show the relationship between the optimal selling rate and the state variables.

In this thesis we discuss basket option valuation for jump-diffusion models. We suggest three new approximate pricing methods. The first approximation method is the weighted sum of Rogers and Shi’s lower bound and the conditional second moment adjustments. The second is the asymptotic expansion to approximate the conditional expectation of the stochastic variance associated with the basket value process. The third is the lower bound approximation which is based on the combination of the asymptotic expansion method and Rogers and Shi’s lower bound. We also derive a forward partial integro-differential equation (PIDE) for general asset price processes with stochastic volatilities and stochastic jump compensators. Numerical tests show that the suggested methods are fast and accurate in comparison with Monte Carlo and other methods in most cases.

The recent turbulence in financial markets, of which a famous casualty is the collapse of the Long Term Capital Management hedge fund, has made market liquidity an issue of high concern to investors and risk managers. The latter group in particular realised that financial models, based on the assumption of perfectly liquid markets where investors can trade large amounts of assets without affecting their prices, may fail miserably under the circumstance where market liquidity vanishes. Understanding the robustness and reliability of models used for trading and risk management purposes is therefore crucially important in the risk analysis. Part I of this thesis studies liquidity risk and its measurement via mean reversion jump diffusion processes. An efficient Monte Carlo method is suggested to find approximate VaR and CVaR for all percentiles with one set of samples from the loss distribution, which applies to portfolios of securities as well as single security. Part II investigates the computational efficiency and flexibility of application of the FFT-based option pricing methodologies. First, an empirical testing of alternative twofactor stochastic volatility affine jump-diffusion models is conducted against an extensive S&P 500 index options data set, using a nonlinear ordinary least squares estimation framework. It is then shown how the two-dimensional FFT may be applied for the pricing of spread options, which have payoff functions and exercise regions that are nonlinear in the underlying log-asset prices. Furthermore, a non-affine four-factor stochastic volatility diffusion model is considered and an approximate CCF specification derived.